which diminishes to zero when \(\xi _t=0, \forall t\) . This manifests the advantage of this doubly-adaptive dual step which creates a bound similar to those in OCO problems without budget constraints, see [1]}. Furthermore, the step is simplified to \(a_t\!=\!a/t^\beta \)  which remains bounded. Hence \(\mathcal {R}_T\!=\!B_T\!=\!\mathcal {O}(1)\) , and if we substitute \(B_T\)  in \(\mathcal {V}_T\) , we get:
\(\mathcal {V}_T\le \sqrt{\frac{-2\mathcal {R}_T}{a_{T-1}}}\le \sqrt{ \frac{-2\mathcal {R}_TT^{\beta }}{a} } = \mathcal {O}\left(T^{\frac{1+\beta }{2}}\right) \) 
